integration with U substitution?

Question

I have answered the first 7 questions but I am confused on this one.
Evaluate the indefinite integral (arctan(2x)) / (1+4x^2)

which one do I substitute u for.

Answer 1

If we choose u=arctan(2x)
du= 2/(4x^2+1)
We are looking for du to be 1/(1+4x^2)
so, we should choose u= 1/2arctan(2x)
This will make du = 1/(1+4x^2)
Now just solve your integral and plug in your values.

Answer 2

The derivative of arctan(2x) = 2/(1+4x^2)

Answer 3

∫ [arctan(2x)/(1+4x^2)] dx

let 2x = y

2dx = dy

dx= dy/2

now the integral is

1/2∫arctan(y)/(1+y^2) dy

let arctan(y) = u

differentiating

[1/(1+y^2)] dy = du

now the integral is

1/2∫u du

1/2(u^2/2) = u^2/4 + c

substituting back u = arctany

(1/4)(arctany)^2 + c

substituting back y = 2x

(1/4)(arctan(2x))^2 + c

Answer 4

2x = tan u
thus u = arctan (2x)...

then solve... if required...


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